There are _________ natural numbers between `n^2 and (n + 1)^2`

Options

A. n+1
B. n
C. `2n^2`
D. 2n

Answer

To find the number of natural numbers between \( n^2 \) and \( (n + 1)^2 \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the two squares**:
We start with the two squares: \( n^2 \) and \( (n + 1)^2 \).

2. **Calculate the difference**:
The difference between \( (n + 1)^2 \) and \( n^2 \) can be calculated as follows:
\[ (n + 1)^2 - n^2 \]

3. **Expand \( (n + 1)^2 \)**:
Using the formula for the square of a binomial:
\[ (n + 1)^2 = n^2 + 2n + 1 \]

4. **Subtract \( n^2 \)**:
Now, substituting this back into our equation:
\[ (n^2 + 2n + 1) - n^2 = 2n + 1 \]

5. **Exclude the endpoints**:
The natural numbers between \( n^2 \) and \( (n + 1)^2 \) do not include \( n^2 \) itself. Therefore, we need to subtract 1 from our result:
\[ (2n + 1) - 1 = 2n \]

6. **Final answer**:
Thus, the total number of natural numbers between \( n^2 \) and \( (n + 1)^2 \) is:
\[ 2n \]

### Final Answer:
There are \( 2n \) natural numbers between \( n^2 \) and \( (n + 1)^2 \).

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